This is a specific continuation of the question of embedding the group of isometries = Iso(M) of a riemannian manifold in $M$ x $(TM)^n$
Having proved the naturality of the exponential map, that is, given isometry $f: M \rightarrow M$ then $f \circ exp_p = exp_{f(p)} \circ df_p$
I can subsequently prove the following if isometry $f:M\to M$ has a fixed point $p$ with $df_p=\text{id} \Rightarrow f=\text{id}$
This proof has the following as a direct Corollary: if I have two isometries $f,\widetilde{f}\colon M \to M'$ between (connected) Riemannian manifolds then if there is $p \in M$ with $f(p) = \widetilde{f}(p)$ and ${\rm d}f_p = {\rm d}\widetilde{f}_p$, we conclude that $f=\widetilde{f}$.
Now as per the Answer by Lee Mosher here: Embedding the group of Isometries of a Riemannian manifold to prove that Iso(M,g) is a finite dimensional lie group
I have a map (which I wish to prove is an embedding) $$F : \text{Iso}(M,g) \to M \times (TM)^n \ \ \ \ \ \ where \ n = dim\ M $$ choose $p \in M$, choose a basis of its tangent space $v_1,...,v_n \in T_p M$, and define $$F(p) = (f(p), df_p(v_1), \ldots, df_p(v_n)) $$
(I was wondering if the notation $F_p$ would be more apt as it acts on isometry group element $f$ and so we get $F_p(f) = (f(p), df_p(v_1), \ldots, df_p(v_n))$ , but that's a minor matter)
I can show that this mapping is injective, as if we have two isometries $f$ and $\tilde{f}$ with $(f(p), df_p(v_1), \ldots, df_p(v_n))$ = $(\tilde{f}(p), {\rm d}\widetilde{f}_p(v_1), \ldots, {\rm d}\widetilde{f}_p(v_n))$, then they are component wise equal and thus $f(p)$ = $\tilde{f}(p)$ and ${\rm d}f_p = {\rm d}\widetilde{f}_p$, which in turn imples that $f$ = $\tilde{f}$ by the above corollary.
Furthermore, since riemannian manifold is locally compact (due to local homeomorphims with $\mathbb R^n$) I can reference the following theorem to give Iso(M) a topology:
(van Dantzig, van der Waerden 1928): If (M,g) is a connected locally compact metric space, then the group Isom(M) of all isometries of M with respect to g is locally compact considered with the compact-open topology
My questions thus are:-
Q1) I cannot quite understand the above theorem and this topology on Iso(M). Subsequently, my trouble now is in showing that this map F is a homeomorphism from Iso(M) to $M$ x $(TM)^n$ onto its image.
Q2) Assuming F is proven to be a homeomorphism, does this finally conclude the proof that it is an embedding (because we would need the differential of F, ie, $F_*$ to be injective as per definition of an immersion) and thus Iso(M) is a finite dimensonal Lie group?
